Your Mac mini likely doesn't have the correct NTFS driver needed for writing to the Seagate Slim external drive. Therefore, to enable write access to it, you'll need to install a reliable third-party NTFS for Mac tool on your Mac.
You can try iBoysoft NTFS for Mac. It's one of the fastest NTFS drivers on the market, with support for macOS 13 Ventura to macOS 10.13 High Sierra. You can use it to write, edit, delete, and copy files on Microsoft NTFS-formatted drives from Mac effortlessly.
It will also automount your NTFS drives without requiring you to relaunch the software after a reboot. Click the button below to download it and get the free trial. Test it out to see if it meets your needs.
If you'd rather work with a file system that's compatible with both macOS and Windows, you can also reformat the external hard drive on Mac to exFAT.